The Lowest Eigenvalue of Schrödinger Operators on Compact Manifolds

The lowest eigenvalue of the Schrödinger operator −Δ+V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$-{\Delta }+\mathcal {V}$\end{document} on a compact Riemannian manifold without boundary is studied. We focus on the particularly subtle case of a sign changing potential with positive average.